On the geodesic distance and group actions on trees

1991 ◽  
Vol 110 (1) ◽  
pp. 67-70
Author(s):  
Marco Pavone
Keyword(s):  
Distance Function ◽  
Geodesic Distance ◽  
Group Actions ◽  
Length Function ◽  
Definite Function ◽  
Negative Definite Function ◽  
Negative Type

The natural distance function on the vertices of a tree is a kernel of negative type. As a corollary, for any group G acting on a tree X, the length function |g| = d(υ, gυ) is a negative definite function on G for any given vertex υ of X.

Parallel Residues

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Distance Function ◽  
Length Function ◽  
Coxeter System ◽  
Vertex Set ◽  
Ordered Pairs

This chapter considers the notion of parallel residues in a building. It begins with the assumption that Δ‎ is a building of type Π‎, which is arbitrary except in a few places where it is explicitly assumed to be spherical. Δ‎ is not assumed to be thick. The chapter then elaborates on a hypothesis which states that S is the vertex set of Π‎, (W, S) is the corresponding Coxeter system, d is the W-distance function on the set of ordered pairs of chambers of Δ‎, and ℓ is the length function on (W, S). It also presents a notation in which the type of a residue R is denoted by Typ(R) and concludes with the condition that residues R and T of a building will be called parallel if R = projR(T) and T = projT(R).

scholarly journals Application of the Lyapunov Algorithm to Optimize the Control Strategy of Low-Voltage and High-Current Synchronous DC Generator Systems

Electronics ◽  
2019 ◽  
Vol 8 (8) ◽  
pp. 871 ◽  
Author(s):  
Jinfeng Liu ◽  
Xiaohai Tan ◽  
Xudong Wang ◽  
Herbert Ho-Ching IU
Keyword(s):  
Power Supply ◽  
Low Voltage ◽  
Definite Function ◽  
High Current ◽  
Generation System ◽  
Control Variables ◽  
Loop Control ◽  
Decoupling Method ◽  
Negative Definite Function ◽  
Current Sharing

In the present study, a novel multiple three-phase low-voltage and high-current permanent magnet synchronous generation system is proposed, which has only half-turn coils per phase. The proposed system is composed of a generator and two confluence plates with 108 rectifier modules. The output can reach up to 10,000 A continuous DC power supply, which is suitable for the outdoors and non-commercial power supply. The application of the Lyapunov algorithm in the synchronous rectification control was optimized. A current sharing loop control was added to the closed-loop control to ensure a stable output voltage and the output current sharing of each rectifier module. Since the two control variables solved by the Lyapunov algorithm were coupled and the negative definite function of the Lyapunov algorithm could not be guaranteed in this system, a simple decoupling method was used to decouple the control variables. Compared to the conventional control, the proposed strategy highly improved the dynamic performance of the system. The effectiveness of the proposed strategy was verified by the simulation. The 5 V/10,000 A hardware experiment platform was built, which proved the feasibility and validity of the proposed strategy for a high-power generation system.

Unbounded Negative Definite Functions

1981 ◽  
Vol 33 (4) ◽  
pp. 862-871 ◽  
Author(s):  
Charles A. Akemann ◽  
Martin E. Walter
Keyword(s):  
Compact Group ◽  
Definite Function ◽  
Compact Set ◽  
Locally Compact ◽  
Negative Definite Function ◽  
Property T ◽  
Strong Dual

Negative definite functions (all definitions are given in § 1 below) on a locally compact, σ-compact group G have been used in several different contexts recently [2, 5, 7, 11]. In this paper we show how such functions relate to other properties such a group may have. Here are six properties which G might have. They are grouped into three pairs with one property of each pair involving negative definite functions. We show that the paired properties are equivalent and, where possible, give counter-examples to other equivalences. We assume throughout that G is not compact.(1A) G does not have property T.(IB) There is a continuous, negative definite function on G which is unbounded.(2A) G has the (weak and/or strong) dual R-L property.(2B) For every closed, non-compact set Q ⊂ G there is a continuous, negative definite function on G which is unbounded on Q.

scholarly journals Invariants and examples of group actions on trees and length functions

1991 ◽  
Vol 34 (2) ◽  
pp. 313-320 ◽  
Author(s):  
David L. Wilkens
Keyword(s):  
Normal Subgroup ◽  
Group Actions ◽  
Length Function ◽  
Length Functions

An action of a group G on a tree, and an associated Lyndon length function l, give rise to a hyperbolic length function L and a normal subgroup K having bounded action. The Theorem in Section 1 shows that for two Lyndon length functions l, l′ to arise from the same action of G on some tree, L = L′ and K = K′. Moreover for L non-abelian L = L′ implies K = K′. That this is not so for abelian L is shown in Section 2 where two examples of Lyndon length functions l, l′ on an H.N.N. group are given, with their associated actions on trees, for which L = L′ is abelian but K≠K′.

Stability Criterion of Discrete Hopfield Neural Networks with Weighted Function Matrix and one-Delay

2011 ◽  
Vol 271-273 ◽  
pp. 725-729
Author(s):  
Hai Yang Zou ◽  
Sha Sha Chen ◽  
Xiao Feng Lai
Keyword(s):  
Neural Networks ◽  
Stability Criterion ◽  
Hopfield Neural Networks ◽  
Definite Function ◽  
Weighted Function ◽  
Negative Definite Function ◽  
Parallel Mode ◽  
Diagonally Dominant ◽  
Serial Mode ◽  
Function Matrix

In this paper, discrete Hopfield neural networks with weight function matrix(WFM) and delay(DHNNWFMD) are presented. And we obtain an important result that if the WFM is a symmetric function matrix(SFM) and delayed function matrix(DFM) is a diagonally dominant function matrix(DDFM), DHNNWFMD will converge to a state in serial mode and if the WFM is a SFM and non-negative definite function matrix(NFM) and DFM is a DDFM, DHNNWFMD will converge to a state in parallel mode. It provides some theory basis for the application of DHNNWFMD.

Minimal group actions on Λ-trees

1998 ◽  
Vol 128 (1) ◽  
pp. 23-36 ◽  
Author(s):  
I. M. Chiswell
Keyword(s):  
Existence And Uniqueness ◽  
Group Actions ◽  
Length Function ◽  
Finitely Generated ◽  
Minimal Action ◽  
Minimal Group ◽  
Finitely Generated Group ◽  
Main Emphasis

We consider the existence and uniqueness of minimal invariant subtrees for abelian actions of groups on Λ-trees, and whether or not a minimal action is determined up to isomorphism by the hyperbolic length function. The main emphasis is on actions of end type. For a trivial action of end type, there is no minimal invariant subtree. However, if a finitely generated group has an action of end type, the action is nontrivial and there is a unique minimal invariant subtree. There are examples of infinitely generated groups with a nontrivial action of end type for which there is no minimal invariant subtree. These results can be used to study actions of cut type.

scholarly journals Remarks on the paper “Transient Markov convolution semi-groups and the associated negative definite functions”

1986 ◽  
Vol 102 ◽  
pp. 181-184 ◽  
Author(s):  
Masayuki Itô
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Definite Function ◽  
Semi Group ◽  
Locally Compact ◽  
Negative Definite Function

Let X be a locally compact and σ-compact abelian group and let denote the dual group of X. We denote by ξ a fixed Haar measure on X and by the Haar measure associated with ξ. In [2], we show the followingTheorem. Let (αt)t≧0 be a sub-Markov convolution semi-group on X and let ψ be the negative definite function associated with (αt)t≧0. Then (αt)t≧0 is transient if and only if Re (1/ψ) is locally -summable.

scholarly journals Non-symmetric translation invariant Dirichlet forms on hypergroups

1987 ◽  
Vol 36 (1) ◽  
pp. 61-72 ◽  
Author(s):  
Walter R. Bloom ◽  
Herbert Heyer
Keyword(s):  
Dirichlet Form ◽  
Dirichlet Forms ◽  
Definite Function ◽  
Two Dimensional ◽  
Gelfand Pairs ◽  
Compact Type ◽  
Translation Invariant ◽  
Negative Definite Function ◽  
Commutative Hypergroup

In this note translation-invariant Dirichlet forms on a commutative hypergroup are studied. The main theorem gives a characterisation of an invariant Dirichlet form in terms of the negative definite function associated with it. As an illustration constructions of potentials arising from invariant Dirichlet forms are given. The examples of one- and two-dimensional Jacobi hypergroups yield specifications of invariant Dirichlet forms, particularly in the case of Gelfand pairs of compact type.

A New Approach to Stable Optimal Control of Complex Nonlinear Dynamical Systems

2013 ◽  
Vol 81 (3) ◽  
Author(s):  
Firdaus E. Udwadia
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Nonlinear Differential Equations ◽  
Nonlinear Dynamical System ◽  
Rate Of Change ◽  
Definite Function ◽  
Constrained Motion ◽  
Nonlinear Controller ◽  
Control Cost ◽  
Nonlinear Dynamical ◽  
Negative Definite Function

This paper gives a simple approach to designing a controller that minimizes a user-specified control cost for a mechanical system while ensuring that the control is stable. For a user-given Lyapunov function, the method ensures that its time rate of change is negative and equals a user specified negative definite function. Thus a closed-form, optimal, nonlinear controller is obtained that minimizes a desired control cost at each instant of time and is guaranteed to be Lyapunov stable. The complete nonlinear dynamical system is handled with no approximations/linearizations, and no a priori structure is imposed on the nature of the controller. The methodology is developed here for systems modeled by second-order, nonautonomous, nonlinear, differential equations. The approach relies on some recent fundamental results in analytical dynamics and uses ideas from the theory of constrained motion.

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